A study of numerical methods for hyperbolic conservation laws with stiff source terms

Abstract

The proper modeling of nonequilibrium gas dynamics is required in certain regimes of hypersonic flow. For inviscid flow this gives a system of conservation laws coupled with source terms representing the chemistry. Often a wide range of time scales is present in the problem, leading to numerical difficulties as in stiff systems of ordinary differential equations. Stability can be achieved by using implicit methods, but other numerical difficulties are observed. The behavior of typical numerical methods on a model advection equation with a parameter-dependent source term is studied. Two approaches to incorporate the source terms are utilized: MacCormack type predictor-corrector methods with flux limiters and splitting methods in which the fluid dynamics and chemistry are handled in separate steps. Comparisons over a wide range of parameter values are made. On the whole, the splitting methods perform somewhat better. In the stiff case, a numerical phenomenon of incorrect propagation speeds of discontinuities is observed and explained. Similar behavior was reported by Colella, Majda, and Roytburd ( SIAM J. Sci. Stat. Comput. 7, 1059 (1986)) on a model combustion problem. Using the model scalar equation, we show that this is due to the introduction of nonequilibrium values through numerical dissipation in the advection step.